Pair sequence number
Pair sequence number is a number calculated with a program that posted at googology thread of Japanese BBS 2ch.net in 2014.First version of BASIC programModified version of BASIC programName of the system and the numberBlog post which introduces pair sequence system The algorithm of the program is called pair sequence system. It is supposed to calculate \(f_{\vartheta(\Omega_\omega)+1}(10)\), and this program is an extention of a program written by the same author which calculates \(f_{\epsilon_0+1}(10)\).BASIC program of \(\epsilon_0\) level and a blog post which introduces it Pair sequence is a finite sequence of pair of nonnegative integers, such as (0,0)(1,1)(2,2)(3,3)(3,2). Pair sequence P works as a function from natural numbers to natural numbers, and it can be denoted as Pn, such as (0,0)(1,1)(2,2)(3,3)(3,2)n. The function Pn can be approximated with Hardy hierarchy, and the pair sequence itself represents the ordinal of the approximated Hardy hierarchy, such as \((0,0)(1,1)(2,2)(3,3)(3,2) = \psi(\psi_1(\Omega_2))\). Original BASIC Code In the loop of "for D=0 to 9" to "next", \(C=f_{\vartheta(\Omega_\omega)}©\) is calculated. Repetition of this loop 10 times results in the pair sequence number of \(f_{\vartheta(\Omega_\omega)+1}(10)\). dim A(Infinity):dim B(Infinity):C=9 for D=0 to 9 for E=0 to C A(E)=E:B(E)=E next for F=C to 0 step -1 C=C*C if B(F)=0 then G=1 else G=0 for H=0 to F*G if A(F-H) Equivalent C code This code is optimized for short characters with Bignum Bakeoff rule. The pair sequence number is returned from main(). It has 278 characters without whitespaces. #define A af #define B bf #define M = malloc(9), #define W while ( main(f) { int *a M *b M c = 9, d = 10, h, i, k; W d--) { f = h = c + 1; W h--) ah = bh = h; W f--) { c *= c; h = f + 1; W h--) (ah < A && (bh < B || !B) || A + B 0)? (k = B ? A - ah: 0, i = f - h, h = 0): 0; h = f + c * i; a = realloc(a, h); b = realloc(b, h); W f < h) A = af-i + k, B = bf-i, f++; } } return c; } Verification code Bashicu matrix calculatorBashicu matrix calculator can show the calcultion process of pair sequence system. Here is some examples of the calculation of pair sequence. In the calculation, original algorithm was modified so that n=2 is not changed. * (0,0)(1,1) * (0,0)(1,1)(1,1) * (0,0)(1,1)(2,0) * (0,0)(1,1)(2,1) * (0,0)(1,1)(2,2) * (0,0)(1,1)(2,2)(3,3) * (0,0)(1,1)(2,2)(3,3)(4,4) Corresponding ordinals \begin{eqnarray*} (0,0) &=& 1 \\ (0,0)(0,0) &=& 2 \\ (0,0)(0,0)(0,0) &=& 3 \\ (0,0)(1,0) &=& \omega \\ (0,0)(1,0)(0,0)(0,0) &=& \omega+2 \\ (0,0)(1,0)(0,0)(0,1) &=& \omega \cdot 2 \\ (0,0)(1,0)(1,0) &=& \omega^2 \\ (0,0)(1,0)(1,0)(0,0)(0,1) &=& \omega^2+\omega \\ (0,0)(1,0)(2,0) &=& \omega^\omega \\ (0,0)(1,0)(2,0)(3,0) &=& \omega^{\omega^\omega} \\ (0,0)(1,0)(2,0)(3,0)(4,0) &=& \omega^{\omega^{\omega^\omega}} \\ (0,0)(1,1) &=& \epsilon_0 \\ (0,0)(1,1)(1,0)(2,1) &=& \epsilon_0^2 \\ (0,0)(1,1)(1,0)(2,1)(2,0)(3,1) &=& \epsilon_0^{\epsilon_0} \\ (0,0)(1,1)(1,0)(2,1)(2,0)(3,1)(3,0)(4,1) &=& \epsilon_0^{\epsilon_0^2} \\ (0,0)(1,1)(1,0)(2,1)(2,0)(3,1)(3,0)(4,1)(4,0)(5,1) &=& \epsilon_0^{\epsilon_0^{\epsilon_0}} \\ (0,0)(1,1)(1,1) &=& \epsilon_1 = \psi(1) \\ (0,0)(1,1)(2,0) &=& \epsilon_{\omega} = \psi(\omega) \\ (0,0)(1,1)(2,0)(2,0) &=& \epsilon_{\omega^2} = \psi(\omega^2) \\ (0,0)(1,1)(2,0)(3,0) &=& \epsilon_{\omega^\omega} = \psi(\omega^\omega) \\ (0,0)(1,1)(2,0)(3,1) &=& \epsilon_{\epsilon_0} = \psi(\psi(0)) \\ (0,0)(1,1)(2,0)(3,1)(4,0)(5,1) &=& \epsilon_{\epsilon_{\epsilon_0}} = \psi(\psi(\psi(0))) \\ (0,0)(1,1)(2,1) &=& \zeta_0 = \phi(2,0) = \psi(\Omega) \\ (0,0)(1,1)(2,1)(3,1) &=& \Gamma_0 = \phi(1,0,0) = \psi(\Omega^\Omega) \\ (0,0)(1,1)(2,1)(3,1)(4,0) &=& ψ(\Omega^{\Omega \cdot \omega}) \\ (0,0)(1,1)(2,1)(3,1)(4,1) &=& ψ(\Omega^{\Omega \cdot \Omega}) = ψ(\Omega^{\Omega^2}) \\ (0,0)(1,1)(2,1)(3,1)(4,1)(4,0) &=& ψ(\Omega^{\Omega^2 \cdot \omega}) \\ (0,0)(1,1)(2,1)(3,1)(4,1)(4,1) &=& ψ(\Omega^{\Omega^2 \cdot \Omega}) = ψ(\Omega^{\Omega^3}) \\ (0,0)(1,1)(2,1)(3,1)(4,1)(5,0) &=& ψ(\Omega^{\Omega^\omega}) \\ (0,0)(1,1)(2,1)(3,1)(4,1)(5,1) &=& ψ(\Omega^{\Omega^\Omega}) \\ (0,0)(1,1)(2,1)(3,1)(4,1)(5,1)(6,1) &=& ψ(\Omega^{\Omega^{\Omega^2}}) \\ (0,0)(1,1)(2,1)(3,1)(4,1)(5,1)(6,1)(7,1) &=& ψ(\Omega^{\Omega^{\Omega^\Omega}}) \\ (0,0)(1,1)(2,2) &=& \psi(\epsilon_{\Omega+1}) = \psi(\psi_1(0)) \\ (0,0)(1,1)(2,2)(0,0) &=& \psi(\psi_1(0))+1 \\ (0,0)(1,1)(2,2)(1,0) &=& \psi(\psi_1(0)) \omega \\ (0,0)(1,1)(2,2)(2,0) &=& \psi(\psi_1(0) \omega) \\ (0,0)(1,1)(2,2)(3,0) &=& \psi(\psi_1(\omega)) \\ (0,0)(1,1)(2,2)(3,0)(4,0) &=& \psi(\psi_1(\omega^\omega)) \\ (0,0)(1,1)(2,2)(3,0)(4,1) &=& \psi(\psi_1(\psi(0)))=\psi(\psi_1(\epsilon_0)) \\ (0,0)(1,1)(2,2)(3,1) &=& \psi(\psi_1(\Omega)) \\ (0,0)(1,1)(2,2)(3,2) &=& \psi(\psi_1(\Omega_2)) \\ (0,0)(1,1)(2,2)(3,3) &=& \psi(\psi_1(\psi_2(0))) \\ (0,0)(1,1)(2,2)(3,3)(4,4) &=& \psi(\psi_1(\psi_2(\psi_3(0)))) \\ (0,0)(1,1)(2,2)(3,3)...(9,9) &=& \psi(\psi_1(\psi_2(\psi_3(\psi_4(\psi_5(\psi_6(\psi_7(\psi_8(0))))))))) \\ \end{eqnarray*} Sources See also ja:ペア数列数 Category:Numbers Category:Legiattic Array Notation Level Category:Googology in Asia